Pythagoras number

In mathematics, the Pythagoras number or reduced height of a field describes the structure of the set of squares in the field. The Pythagoras number ${\displaystyle p(K)}$ of a field ${\displaystyle K}$ is the smallest positive integer ${\displaystyle p}$ such that every sum of squares in ${\displaystyle K}$ is a sum of ${\displaystyle p}$ squares.

A Pythagorean field is a field with Pythagoras number 1: that is, every sum of squares is already a square.

Examples

• Every non-negative real number is a square, so ${\displaystyle p(\mathbb {R} )=1}$.
• For a finite field of odd characteristic, not every element is a square, but all are the sum of two squares, so ${\displaystyle p=2}$.
• By Lagrange's four-square theorem, every positive rational number is a sum of four squares, and not all are sums of three squares, so ${\displaystyle p(\mathbb {Q} )=4}$.

Properties

• Every positive integer occurs as the Pythagoras number of some formally real field.

• The Pythagoras number is related to the Stufe by ${\displaystyle p(F)\leq s(F)+1}$. If ${\displaystyle F}$ is not formally real then ${\displaystyle s(F)\leq p(F)\leq s(F)+1}$, and both cases are possible: for ${\displaystyle F=\mathbb {C} }$ we have ${\displaystyle s=p=1}$, whereas for ${\displaystyle F=\mathbb {F} _{5}}$ we have ${\displaystyle s=1}$, ${\displaystyle p=2}$.

• As a consequence, the Pythagoras number of a non-formally-real field is either a power of 2, or 1 more than a power of 2. All such cases occur: i.e., for each pair ${\displaystyle (s,p)}$ of the form ${\displaystyle (2^{k},2^{k})}$ or ${\displaystyle (2^{k},2^{k}+1)}$, there exists a field ${\displaystyle F}$ such that ${\displaystyle (s(F),p(F))=(s,p)}$. For example,
  • quadratically closed fields and fields of characteristic 2 give ${\displaystyle (s(F),p(F))=(1,1)}$;
  • for primes ${\displaystyle p\equiv 1\!\!\!\!{\pmod {4}}}$, ${\displaystyle \mathbb {F} _{p}}$ and the p-adic field ${\displaystyle \mathbb {Q} _{p}}$ give ${\displaystyle (1,2)}$;
  • for primes ${\displaystyle p\equiv 3\!\!\!\!{\pmod {4}}}$, ${\displaystyle \mathbb {F} _{p}}$ gives ${\displaystyle (2,2)}$ and ${\displaystyle \mathbb {Q} _{p}}$ gives ${\displaystyle (2,3)}$;
  • ${\displaystyle \mathbb {Q} _{2}}$ gives ${\displaystyle (4,4)}$, and
  • the function field ${\displaystyle \mathbb {Q} _{2}(t)}$ gives ${\displaystyle (4,5)}$.

• The Pythagoras number is related to the height of a field ${\displaystyle F}$: if ${\displaystyle F}$ is formally real then ${\displaystyle h(F)}$ is the smallest power of 2 which is not less than ${\displaystyle p(F)}$; if ${\displaystyle F}$ is not formally real then ${\displaystyle h(F)=2s(F)}$.

Notes

References